Airfoil shape for thirteenth stage compressor rotor blade

ABSTRACT

A system is provided, including an airfoil. The airfoil includes a first suction portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a suction side as set forth in TABLE I to a maximum of three decimal places, wherein the X and Y values of the suction side are coordinate values that couple together to define suction side sections of the first suction portion of the nominal airfoil profile at each Z coordinate value, the suction side sections of the first suction portion of the nominal airfoil profile are coupled together to define the first suction portion, the airfoil includes an airfoil length along a Z axis, the first suction portion comprises a first portion length along the Z axis, the first portion length is less than or equal to the airfoil length, and the Cartesian coordinate values of X, Y, and Z are non-dimensional values convertible to dimensional distances.

BACKGROUND OF THE INVENTION

The subject matter disclosed herein relates to compressor rotor blades,and more specifically, to a thirteenth stage compressor rotor blade.

Compressors are used in a variety of industries and systems to compressa gas, such as air. For example, gas turbine engines typically include acompressor to provide compressed air for combustion and cooling.Compressors typically include a rotor assembly and a stator assembly. Inmulti-stage compressors, the rotor assembly may include multiple rows(e.g., rotor stages) each row having multiple rotor blades. Likewise,the stator assembly may include multiple rows (e.g., stator stages) eachrow having multiple stator vanes. The rotor assembly is designed torotate with respect to the stator assembly, compressing an intake fluidas the fluid traverses the compressor.

BRIEF DESCRIPTION OF THE INVENTION

Certain embodiments commensurate in scope with the originally claimedsubject matter are summarized below. These embodiments are not intendedto limit the scope of the claimed invention, but rather theseembodiments are intended only to provide a brief summary of possibleforms of the subject matter. Indeed, the invention may encompass avariety of forms that may be similar to or different from theembodiments set forth below.

In a first embodiment, a system includes an airfoil. The airfoilincludes a first suction portion of a nominal airfoil profilesubstantially in accordance with Cartesian coordinate values of X, Y,and Z of a suction side as set forth in TABLE I to a maximum of threedecimal places, wherein the X and Y values of the suction side arecoordinate values that couple together to define suction side sectionsof the first suction portion of the nominal airfoil profile at each Zcoordinate value, the suction side sections of the first suction portionof the nominal airfoil profile are coupled together to define the firstsuction portion, the airfoil includes an airfoil length along a Z axis,the first suction portion comprises a first portion length along the Zaxis, the first portion length is less than or equal to the airfoillength, and the Cartesian coordinate values of X, Y, and Z arenon-dimensional values convertible to dimensional distances.

In a second embodiment, a system includes an airfoil. The airfoilincludes a suction side of a nominal airfoil profile substantially inaccordance with Cartesian coordinate values of X, Y, and Z of thesuction side as set forth in TABLE I to a maximum of three decimalplaces, wherein the X and Y values of the suction side are coordinatevalues that couple together to define suction side sections of thesuction side of the nominal airfoil profile at each Z coordinate value,the suction side sections of the suction side of the nominal airfoilprofile are coupled together to define the suction side, and theCartesian coordinate values of X, Y, and Z are non-dimensional valuesconvertible to dimensional distances.

In a third embodiment, a system includes an airfoil. The airfoilincludes a nominal airfoil profile substantially in accordance withCartesian coordinate values of X, Y, and Z as set forth in TABLE I to amaximum of three decimal places, wherein the X and Y values arecoordinate values that couple together to define airfoil sections of thenominal airfoil profile at each Z coordinate value, the airfoil sectionsof the nominal airfoil profile are coupled together to define anentirety of the airfoil, and the Cartesian coordinate values of X, Y,and Z are non-dimensional values convertible to dimensional distances.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features, aspects, and advantages of the present subjectmatter will become better understood when the following detaileddescription is read with reference to the accompanying drawings in whichlike characters represent like parts throughout the drawings, wherein:

FIG. 1 is a block diagram of an embodiment of a gas turbine systemhaving a multi-stage axial compressor;

FIG. 2 is a cross-sectional side view of an embodiment of the gasturbine engine of FIG. 1, illustrating stages of blades and vanes of themulti-stage axial compressor;

FIG. 3 is side view of an embodiment of an airfoil of a compressor rotorblade or a compressor stator vane;

FIG. 4 is side view of an embodiment of an airfoil of a compressor rotorblade or a compressor stator vane;

FIG. 5 is a side view of an embodiment of an airfoil of the compressorrotor blade or a compressor stator vane; and

FIG. 6 is an axial view of an embodiment of the airfoil of thecompressor rotor blade or compressor stator vane of FIGS. 3-5.

DETAILED DESCRIPTION OF THE INVENTION

One or more specific embodiments of the present subject matter will bedescribed below. In an effort to provide a concise description of theseembodiments, all features of an actual implementation may not bedescribed in the specification. It should be appreciated that in thedevelopment of any such actual implementation, as in any engineering ordesign project, numerous implementation-specific decisions must be madeto achieve the developers' specific goals, such as compliance withsystem-related and business-related constraints, which may vary from oneimplementation to another. Moreover, it should be appreciated that sucha development effort might be complex and time consuming, but wouldnevertheless be a routine undertaking of design, fabrication, andmanufacture for those of ordinary skill having the benefit of thisdisclosure.

When introducing elements of various embodiments of the present subjectmatter, the articles “a,” “an,” “the,” and “said” are intended to meanthat there are one or more of the elements. The terms “comprising,”“including,” and “having” are intended to be inclusive and mean thatthere may be additional elements other than the listed elements.

The disclosed embodiments include a multi-stage axial compressor, whichmay be a standalone unit or part of a multi-stage rotary machine such asa gas turbine system. In certain embodiments, the multi-stage axialcompressor includes a plurality of rotor stages and corresponding statorstages, each rotor stage and stator stage including a plurality ofairfoils (e.g., blades or vanes) with a specific airfoil profile. Duringoperation, a compressible fluid (e.g., gas such as air, oxygen, oxygenenriched air, oxygen reduced air, exhaust gas, nitrogen, etc.) may enterthe multi-stage axial compressor through an inlet system, and each stageof the multi-stage axial compressor will generally increase the pressureand temperature of the compressible fluid by a certain amount. In a gasturbine system, a compressed fluid may then be delivered, for example,via an outlet system, to a combustor for combustion with a fuel. Theamount of pressure and temperature increase at each stage of themulti-stage axial compressor may depend on particular operatingconditions, such as speed, inlet boundary conditions (e.g., flow,pressure, temperature, composition, and so forth), outlet boundaryconditions (e.g., flow resistance, and so forth), and stage efficiency.

During compression, an energy level of the compressible fluid mayincrease as the compressible fluid flows through the multi-stage axialcompressor due to the exertion of a torque on the fluid by the rotatingrotor blades. The stator's stationary vanes slow the compressible fluid,converting a circumferential component of the flow into pressure. Anairfoil profile or design of the airfoil (e.g., rotor blades and/orstator vanes) may directly affect compression of the compressible fluid.Airfoil profiles described herein may be more optimized and matched forspecific velocities and turning speeds. Further, the airfoil profilesdescribed herein may be more optimized for specific stages of themulti-stage axial compressor with a specific total number of stages.Additionally, the airfoil profiles described herein may be designed forcompressor rotor blades, compressor stator vanes, or any combinationthereof. In certain embodiments, the airfoil profiles described hereinmay be designed for compressor rotor blades and/or compressor statorvanes in any one or more stages of a multi-stage axial compressor with2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, or 20stages. More specifically, in certain embodiments, the airfoil profilesdescribed herein may be designed for compressor rotor blades and/orcompressor stator vanes in stage 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,13, and/or 14 of a 14 stage axial compressor. For example, in certainembodiments, the airfoil profiles described herein may be more optimizedfor a thirteenth stage airfoil (e.g., a rotor blade or stator vane) of a14 stage axial compressor. The airfoils (e.g., blades or vanes)described herein may be described in terms of X, Y, and Z values setforth in certain table(s) described herein that define airfoil sectionsof the airfoils. In certain embodiments the X, Y, and Z values maydescribe suction side airfoil surfaces, pressure side airfoil surfaces,or a combination thereof. The X, Y, and Z values may include coordinatevalues in a Cartesian coordinate system, including negative and positivevalues denoting sides that are opposite to each other from a certainaxis (e.g., X, Y, Z axis).

Turning now to FIG. 1, the figure is a block diagram of an embodiment ofa turbine system 10 that includes a gas turbine engine 12 and acontroller 14 (e.g., electronic and/or processor-based controller). Thecontroller 14 may be communicatively coupled to a variety of sensors 16and actuators 18 disposed in various components of the gas turbineengine 12. Signals received via the sensors 16 may be used to derivecontrol actions executable via the actuators 18. The illustrated gasturbine engine 12 includes a compressor 20 (e.g., a multi-stage axialcompressor or compressor section), a turbine 22 (e.g., a multi-stageturbine or turbine section), and fuel nozzles 24 coupled to one or morecombustors 26 (e.g., compressor section). The compressor 20 and turbine22 each may have any number of rows stages of rotor blades and statorvanes (e.g., 1 to 20). As discussed in detail below, an embodiment of anairfoil profile is provided for use with airfoils (e.g., rotor blades orstator vanes) in one or more stages of the compressor 20. However,before presenting details of the airfoil profile, the followingdiscussion provides a brief description of the gas turbine engine 12 andits operation.

In operation, the compressor 20 is configured to compress a compressiblefluid (e.g., gas such as air, oxygen, and/or exhaust gas), and deliverthe compressed fluid to the fuel nozzles 24 and/or combustors 26.Although the compressible fluid may include any suitable gas, thefollowing discussion may generally refer to the compressible fluid as anoxidant (e.g., air) as one non-limiting example. The fuel nozzles 24 areconfigured to supply fuel (e.g., from one or more fuel supplies) intothe one or more combustors 26 (e.g., in combustion chambers), whichcombust the fuel with the oxidant (e.g., air) to generate hot combustiongases to drive the turbine 22. The fuel nozzles 24 may be designed aspre-mix fuel nozzles 24 and/or diffusion fuel nozzles 24. Pre-mix fuelnozzles 24 mix the fuel with the oxidant (e.g., air) to create pre-mixtype flames. Diffusion fuel nozzles 24 do not premix the fuel with theoxidant, and thus create diffusion type flames. Regardless of the typeof flames, the hot combustion gas flows from the combustors 26 into theturbine 22, thereby driving rotation of one or more stages of turbineblades coupled to a turbine rotor and shaft 30 along an axis 32.Eventually, the hot combustion gas exits the turbine 22 through anexhaust outlet 28 (e.g., exhaust stack, exhaust end). In the illustratedembodiment, the shaft 30 is coupled to the compressor 20 and a load 36,such that rotation of the shaft 30 also drives rotation of thecompressor 20 and the load 36. The compressor 20 may intake the oxidant(e.g., air) through an air intake 34, which may include filters, thermalcontrol systems, or any other preconditioning systems. The load 36 mayinclude an electrical generator, a rotary machine, a propulsion systemof a vehicle, or any other suitable device.

The airfoil profile described in further detail below may be used in anystage of the compressor 20 (e.g., multi-stage axial compressor with anynumber of stages of rotor blades and stator vanes). The compressor 20may include rotating blades and stationary vanes (e.g., airfoils) thatmay be disposed in rows or stages, described in more detail below. Theoxidant (e.g., air) may be progressively compressed in stages or rows ofrotating blades and corresponding stator vanes as the air movesdownstream in the compressor 20. In the depicted embodiment, thecompressor 20 is a multi-stage axial compressor 20 having at least tworows or stages of blades and vanes. For example, in certain embodiments,the multi-stage axial compressor 20 may have 14 rows or stages ofcompressor blades and vanes.

It may be beneficial to illustrate a more detailed view of certaincomponents of the gas turbine engine 12. Accordingly, FIG. 2 is across-sectional side view of an embodiment of the compressor 20 of thegas turbine engine 12 of FIG. 1. Throughout the discussion of FIG. 2, aset of axes will be referenced. These axes are based on a cylindricalcoordinate system and point in an axial direction 38 (e.g., downstream),a radial direction 40, and a circumferential direction 42. For example,the axial direction 38 extends downstream through the compressor 20generally parallel to the axis 32, the radial direction 40 extends awayfrom the axis 32, and the circumferential direction 42 extends aroundthe axis 32.

In operation, air enters the compressor 20 in the axial direction 38through the air intake 34 and may be pressurized in the multi-stageaxial compressor 20. The compressed air may then be mixed with fuel forcombustion within the combustor 26 to drive the turbine 22 to rotate theshaft 30 in the circumferential direction 42 and, thus, the multi-stageaxial compressor 20 and the load 36. The rotation of the shaft 30 alsocauses one or more blades 44 (e.g., compressor rotor blades) within themulti-stage axial compressor 20 to draw in and pressurize the airreceived by the air intake 34.

The multi-stage axial compressor 20 may include a rotor assembly 46having multiple rotor blades 44 surrounded by a static casing 48 havingmultiple stator vanes 50 (e.g., variable stator vanes and/or fixedstator vanes). In some embodiments, the static casing 48 of thecompressor 20 or the air intake 34 may have one or more sets of inletguide vanes 52 (IGVs) (e.g., variable IGV stator vanes) that may controlflows into the compressor 20. Each variable stator vane 50 (includingeach variable IGV stator vane 52) may be configured to vary its vaneangle relative to the gas flow (e.g. air flow) by rotating the vane 50,52 about an axis of rotation (e.g., radially oriented vane shaft).However, each variable stator vane 50 may be otherwise stationaryrelative to the rotor blades 44. In certain embodiments, each variablestator vane 50 may be coupled to an actuator 18 (e.g., electric drive,pneumatic drive, or hydraulic drive), which is coupled to a controller14 configured to vary the vane angle in response to feedback fromsensors 16. Each fixed stator vane 50 may be configured to remain in afixed angular position, such that the vane angle does not vary. Thecompressor 20 may include a plurality of rows or stages 54, such asbetween 2 to 30, 2 to 25, 2 to 20, 2 to 14, or 2 to 10 rows or stages,or any specific number or range therebetween. In each stage, themulti-stage axial compressor 20 may include 2 to 1000, 5 to 500, or 10to 100 rotor blades 44, and 2 to 1000, 5 to 500, or 10 to 100 statorvanes 50. In particular, the illustrated embodiment of the multi-stageaxial compressor 20 includes 14 stages. It may be appreciated that eachstage 54 has a set of rotor blades 44 disposed at a first axial positionand a set of stator vanes 50 disposed at a second axial position alongthe length of the compressor 20. In other words, each stage 54 has therotor blades 44 and stator vanes 50 axially offset from one another,such that the compressor 20 has an alternating arrangement of rotorblades 44 and stator vanes 50 one set after another along the length ofthe compressor 20. Each set of rotor blades 44 extends (e.g., in aspaced arrangement) in the circumferential direction 42 about the shaft30, and each set of stator vanes 50 extends (e.g., in a spacedarrangement) in the circumferential direction 42 within the staticcasing 48. While the compressor 20 may include greater or fewer stages54 than 14, FIG. 2 illustrates an embodiment of the compressor 20 with14 stages 54 identified as follows: first stage 54 a, second stage 54 b,third stage 54 c, fourth stage 54 d, fifth stage 54 e, sixth stage 54 f,seventh stage 54 g, eighth stage 54 h, ninth stage 54 i, tenth stage 54j, eleventh stage 54 k, twelfth stage 54 l, thirteenth stage 54 m, andfourteenth stage 54 n. In certain embodiments, each stage 54 may includerotor blades 44 and stator vanes 50 (e.g., fixed stator vanes 50 and/orvariable stator vanes 50). For example, in certain embodiments, earlierstages 54 (e.g., 54 a, 54 b, 54 c, etc.) may include variable statorvanes 50, while later stages 54 may include fixed stator vanes 50.

The airfoil described in the TABLE I below may describe either a rotorblade 44 or a stator vane 50 of the compressor 20. For example, theairfoil described in the TABLE I below may be placed as a rotor blade 44of the thirteenth stage 54 m. In use, the rotor blades 44 may rotatecircumferentially about the static casing 48 and the stator vanes 50.Rotation of the rotor blades 44 may result in air entering the airintake 34. The air is then subsequently compressed as it traverses thevarious stages 54 (e.g., first stage 54 a to fourteenth stage 54 n) ofthe compressor 20 and moves in the axial direction 38 downstream of themulti-stage axial compressor 20. The compressed air may then exitthrough an outlet 56 of the multi-stage axial compressor 20. The outlet56 may have a set of exit guide vanes 58 (EGVs). The compressed air thatexits the compressor 20 may be mixed with fuel, directed to thecombustor 26, directed to the turbine 22, or elsewhere in the turbinesystem 10.

Certain designs of the rotor blades 44 and stator vanes 50 (e.g.,airfoils) provide for a more efficient multi-stage axial compressor 20system. For example, certain rotor blade and/or stator vane (e.g.,airfoil) designs may improve compressor 20 efficiency and enableimproved operations for the turbine system 10. Referring now to FIG. 3,the figure is a side view of an embodiment of an airfoil 82 that may beincluded in the multi-stage axial compressor 20 as a rotor blade 44 orstator vane 50. In the particular embodiment, the airfoil 82 may beincluded in the thirteenth stage 54 m of the multi-stage axialcompressor 20 as a rotor blade 44. In the depicted embodiment, theairfoil 82 is disposed on a base 60, which may, in certain embodiments,be removably coupled as a rotor blade 44 to the rotor assembly 46. Thatis, the base 60 having the airfoil 82 may be removed from the rotorassembly 46, for example, to inspect, repair, and/or replace the airfoil82. Additionally, or in the alternative, the airfoil 82 may be removablycoupled as a stator vane 50 to the static casing 48. That is, the base60 having the airfoil 82 may be removed from the static casing 48, forexample, to inspect, repair, and/or replace the airfoil 82. The base 60may include a removable mount or coupling 59, such as a dovetail joint.For example, the coupling 59 may include a T-shaped structure, a hook,one or more lateral protrusions, one or more lateral slots, or anycombination thereof. The coupling 59 (e.g., dovetail joint) may beconfigured to mount into the rotor assembly 46 or the static casing 48in an axial direction 38, a radial direction 40, and/or acircumferential direction 42 (e.g., into an axial slot or opening, aradial slot or opening, and/or a circumferential slot or opening).

As further described herein, the airfoil 82 includes a suction side 62and a pressure side 64 disposed opposite from one another on the airfoil82 (i.e., opposite faces). The airfoil 82 also includes leading andtrailing edges 61 and 63 disposed opposite from one another on theairfoil 82 (e.g., opposite upstream and downstream edges). The suctionside 62, the pressure side 64, the leading edge 61, and the trailingedge 63 generally extend from the base 60 to a tip 68 of the airfoil 82.The leading and trailing edges, 61 and 63 respectively, may be describedas the dividing or intersecting lines between the suction side 62 andthe pressure side 64. In other words, the suction side 62 and thepressure sides 64 couple together with one another along the leadingedge 61 and the trailing edge 63, thereby defining an airfoil shapedcross-section that gradually changes lengthwise along the airfoil 82.The airfoil profile described in further detail below may be utilizedalong any portion or the entirety of the airfoil 82 between the base 60and the tip 68. For example, the portion having the disclosed airfoilprofile may include all or part of the suction side 62, all or part ofthe pressure side 64, or a combination thereof.

In operation, the rotor blades 44 rotate about an axis 66 exerting atorque on a working fluid, such as air, thus increasing energy levels ofthe fluid as the working fluid traverses the various stages 54 of themulti-stage axial compressor 20 on its way to the combustor 26. Thesuction side 62 creates and/or increases a suction force on the workingfluid, while the pressure side 64 creates and/or increases a pressurebias on the working fluid. The rotor blades 44 may be adjacent (e.g.,upstream and/or downstream) to the one or more stationary stator vanes50. The stator vanes 50 slow the working fluid during rotation of therotor blades 44, converting a circumferential component of movement ofthe working fluid flow into pressure. Accordingly, continuous rotationof the rotor blade 44 creates a continuous flow of compressed workingfluid, suitable for combustion via the combustor 26.

The airfoil 82 (e.g., rotor blade 44, stator vane 50) includes anairfoil length L measured from the tip 68 of the airfoil 82 to a bottomregion 70 of the airfoil 82 adjacent the base 60 (e.g., at anintersection of the airfoil 82 with the base 60). An X axis 72 liesparallel to the base 60 and to the rotational axis 66. The rotationalaxis 66 may be parallel to the axis 32 or the shaft 30. The X axis 72 isorthogonal to a Z axis 74 which bisects the airfoil 82. A Y axis 76(shown coming out of the plane of the drawing) is orthogonal to both theX axis 72 and the Z axis 74. The X axis 72 and the Y axis 76 may be usedto define an airfoil profile, shape, or section, for example, takenthrough line 6-6 at a point along the Z axis 74. That is, the airfoilprofile may include an outline of the surface (e.g., section) of theairfoil 82 (e.g., rotor blade 44, stator vane 50) at a point along the Zaxis 74. The airfoil profile may include X, Y, and Z values for thesuction side 62, and X, Y, and Z values for the pressure side 64. ACartesian coordinate system point 78 (e.g., origin) may be used todefine a zero point for the X axis 72, the Z axis 74, and the Y axis 76of the respective airfoil 82. TABLE I below lists variousnon-dimensionalized airfoil shapes for the suction side 62 and thecorresponding pressure side 64 disposed at locations along the Z axis 74from the bottom region 70 to the tip 68 of the airfoil 82.

The airfoil 82 may be described in terms of certain airfoil sectionscontaining various air foil shapes and corresponding rows of the TABLEI. For example, as illustrated in FIG. 4, the airfoil 82 may bedescribed via airfoil shapes disposed on one or more portions 80. Incertain embodiments, the portion 80 of the airfoil 82 may be describedas an area of interest, an area of greater importance, or a sweet spot,wherein the particular airfoil profile may have a greater impact on theperformance, efficiency, and other attributes of the airfoil 82 ascompared with other areas of the airfoil 82. However, the portion 80 mayinclude any area of the airfoil 82, regardless of importance. The one ormore portions 80 may include a suction side portion of the suction side62, a pressure side portion of the pressure side 64, or any combinationthereof. For example, the one or more portions 80 may include suctionside portions 62 and pressure side portions 64 that are offset from oneanother without any overlap along the Z axis 74, suction side portions62 and pressure side portions 64 that partially overlap along the Z axis74, or suction side portions 62 and pressure side portions 64 thatcompletely overlap along the Z axis 74. Portion 80 is shown as arectangle in dashed lines. More specifically, FIG. 4 is a side view ofan embodiment of the airfoil 82 illustrating the portion 80. Asdescribed above, the airfoil 82 may be the rotor blade 44 or the statorvane 50, such as the rotor blade 44 of the thirteenth stage 54 m.Because the figure depicts like elements to FIG. 3, the like elementsare illustrated with like numbers. In the depicted embodiment, theairfoil 82 includes the length L (e.g., total length), as mentionedpreviously, measured along the Z axis 74 (e.g., in the radial direction40) from the tip 68 of the airfoil 82 to the bottom region 70 of theairfoil 82.

The portion 80 may begin at a distance or position d and include alength l extending away from the base 60 in the Z direction along the Zaxis 74. As appreciated, in embodiments having one or more suction sideportions 80 on the suction side 62 and/or one or more pressure sideportions 80 on the pressure side 64, each portion 80 may be defined by alength l and a position d. A zero value of the position d corresponds tothe bottom region 70 of the airfoil 82 adjacent the base 60 (e.g., at anintersection of the airfoil 82 with the base 60), which also correspondsto the coordinate origin 78. When d is zero and l is equal to L, theportion 80 includes the entirety of the airfoil 82 from the bottomregion 70 to the tip 68 of the airfoil 82. By varying values for d andl, portions 80 having varying lengths and start locations from thecoordinate origin 78 may be provided for defining the area of interest(e.g., sweet spot) along the airfoil 82. Each portion 80 may include oneor more adjacent airfoil shapes (e.g., airfoil sections or airfoilshapes 110) “stacked” on top of each other along the Z axis 74,described in more detail below with respect to FIG. 6 and TABLE I below.Each airfoil section or airfoil shape 110 corresponds to Cartesiancoordinate values of X, Y, and Z for a common Cartesian coordinate valueof Z in TABLE I. Furthermore, adjacent airfoil sections or airfoilshapes 110 correspond to the Cartesian coordinate values of X, Y, and Zfor adjacent Cartesian coordinate values of Z in the TABLE I.

With reference to TABLE I, the position d may be used to define a startposition (e.g., first Cartesian coordinate value of Z) of the portion 80in the Z direction along the Z axis 74, while a sum of the position dand the length l may be used to define an end position (e.g., lastCartesian coordinate value of Z) of the portion 80 in the Z directionalong the Z axis 74. In certain embodiments, the position d (e.g., startposition) may be selected directly from one of the Cartesian coordinatevalues of Z in TABLE I, and the sum of the position d and the length l(e.g., end position) may be selected directly from one of the Cartesiancoordinate values of Z in TABLE I. In other embodiments, the desiredvalues of the position d and the length l may be initially selectedwithout referencing TABLE I, and then TABLE I may be subsequentlyanalyzed to select best fits of the Cartesian coordinate values of Z inTABLE I based on the desired values of d and l. For example, withreference to TABLE I, the start position of the portion 80 maycorrespond to the Cartesian coordinate value of Z equal to or nearest tothe value of the position d (e.g., start Z value). If the position d ismidway between adjacent Cartesian coordinate values of Z in TABLE I,then the lesser or greater Cartesian coordinate value of Z may beselected for the start position of the portion 80 (e.g., start Z value).Alternatively, in some embodiments, if a specific value of the positiond is desired but does not match the specific Cartesian coordinate valuesof Z in TABLE I, then regression analysis and/or curve fitting may beused to analyze the data in TABLE I and interpolate new Cartesiancoordinate values of X, Y, Z to enable use of the desired d value.Similarly, with reference to TABLE I, the end position (e.g., end Zvalue) may correspond to the Cartesian coordinate value of Z equal to ornearest to the sum of the position d and the length l. If the sum of theposition d and the length l is midway between adjacent Cartesiancoordinate values of Z in TABLE I, then the lesser or greater Cartesiancoordinate value of Z may be selected for the end position (e.g., end Zvalue). Alternatively, in some embodiments, if a specific value of thelength l is desired but the sum of the position d and the length l doesnot match the specific Cartesian coordinate values of Z in TABLE I, thenregression analysis and/or curve fitting may be used to analyze the datain TABLE I and interpolate new Cartesian coordinate values of X, Y, Z toenable use of the desired l value.

In certain embodiments, the portion 80 may be defined by the Cartesiancoordinate values of X, Y, and Z corresponding to the start Z value, theend Z value, and all intermediate Z values in TABLE I. However, in someembodiments, if the Z values do not match the desired start and endpositions, then the portion 80 may be defined by the Cartesiancoordinate values of X, Y, and Z in TABLE I in the Z direction betweenthe start and end positions (e.g., based on the position d and lengthl). Furthermore, as discussed herein, the portion 80 may include theCartesian coordinate values of X, Y, and Z for the suction side 62(e.g., suction side profile 112—see FIG. 6), the pressure side 64 (e.g.,pressure side profile—see FIG. 6), or a combination thereof.

In certain embodiments, the portion 80 may include the airfoil profileof TABLE I only for the suction side 62 according to the position d andlength l, only for the pressure side 64 according to the position d andlength l, or for both the suction and pressure sides 62 and 64 accordingto the position d and length l. The position d of the portion 80 may begreater than or equal to approximately 0, 5, 10, 15, 20, 25, 30, 35, 40,45, 50, 55, 60, 65, 70, or 75 percent of the length L of the airfoil 82.Furthermore, the length l of the portion 80 may be greater than or equalto approximately 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70,75, 80, 85, 90, 95, or 100 percent of the length L of the airfoil 82.For example, the portion 80 selected from TABLE I may be the suctionside 62 of the outer third of the airfoil 82. In another example, theportion 80 selected from TABLE I may be both the suction side 62 and thepressure side 64 of an interior portion 80 of the airfoil 82, where d isgreater than 0 and l is less than L (e.g., l=0.3L).

Additionally or alternatively, a portion of the airfoil 82, such asportion 80, may be described in terms of a start height (e.g., firstposition) and a stop height (e.g., second position) along the Z axis 74(e.g., in the radial direction 40) as illustrated in FIG. 5.Collectively, the start and stop heights (e.g., first and secondpositions) define a range along the Z axis 74. Again, the portion (e.g.,80) of the airfoil 82 may be described as an area of interest, an areaof greater importance, or a sweet spot, wherein the particular airfoilprofile may have a greater impact on the performance, efficiency, andother attributes of the airfoil 82 as compared with other areas of theairfoil 82. However, the portion (e.g., 80) may include any area of theairfoil 82, regardless of importance. For example, as shown in FIG. 5,an embodiment of the airfoil 82 is illustrated with multiple definedheights 90, 92, 94, 96, 98, and 100 along the Z axis 74. The heights 90,92, 94, 96, 98, and 100 may divide the airfoil 82 into a plurality ofportions along the Z axis 74, wherein the plurality of portions may haveequal lengths along the Z axis 74 (e.g., five portions each being 20percent of the total length L of the airfoil 82). However, in someembodiments, the plurality of portions defined by the heights 90, 92,94, 96, 98, and 100 may have different lengths along the Z axis 74.Although the illustrated embodiment includes 6 heights (e.g., 90, 92,94, 96, 98, and 100) defining 5 portions, certain embodiments mayinclude any number of heights (e.g., 2 to 100 or more) defining anynumber of portions (e.g., 2 to 100 or more) with equal or differentlengths along the Z axis 74.

For example, certain embodiments may include 11 heights to define 10portions with equal or different lengths along the Z axis 74 (e.g., 10portions each being 10 percent of the total length L of the airfoil 82).By further example, certain embodiments may include 101 heights todefine 100 portions with equal or different lengths along the Z axis 74(e.g., 100 portions each being 1 percent of the total length L of theairfoil 82). In this manner, the heights may be used to define aspecific portion (e.g., 80) of the airfoil 82, wherein the specificportion (e.g., area of interest) may track the airfoil profile describedbelow in TABLE I. Again, similar to the discussion of FIG. 4, thespecific portion (e.g., 80) defined by the heights in FIG. 5 may trackthe airfoil profile of TABLE I along only the suction side 62, only thepressure side 64, or both the suction and pressure sides 62 and 64.Because FIG. 5 depicts like elements to FIG. 4, the like elements areillustrated with like numbers.

In the illustrated embodiment, the portion (e.g., 80) may be defined bythe start height and the stop height, wherein the start height is closerto the bottom region 70 than the stop height, and each of the start andstop heights may be selected from one of the heights 90, 92, 94, 96, 98,or 100. Multiple start and stop heights 90, 92, 94, 96, 98, 100 may bedefined to divide the airfoil 82 and define the portion (e.g., 80). Forexample, a first start height 92 and a first stop height 90 may describea first section or segment 102 (e.g., portion 80) of the airfoil 82, asecond start height 96 and a second stop height 92 may describe a secondsection or segment 104 (e.g., portion 80) of the airfoil 82, and a thirdstart height 96 and a third stop height 94 may describe a third sectionor segment 106 (e.g., portion 80) of the airfoil 82. As shown in FIG. 5,each height 90, 92, 94, 96, 98, 100 may define a start height for asegment, a stop height for the segment, or a start height for onesegment and a stop height for another segment. Each of the heights 90,92, 94, 96, 98, 100 may be selected to correspond to a differentCartesian coordinate value of Z in TABLE I. In certain embodiments, eachof the heights 90, 92, 94, 96, 98, 100 may be selected directly from oneof the Cartesian coordinate values of Z in TABLE I. In otherembodiments, the desired values of the heights 90, 92, 94, 96, 98, 100may be initially selected without referencing TABLE I, and then TABLE Imay be subsequently analyzed to select best fits of the Cartesiancoordinate values of Z in TABLE I based on the desired values of theheights 90, 92, 94, 96, 98, 100. For example, each of the heights 90,92, 94, 96, 98, 100 may correspond to the Cartesian coordinate value ofZ equal to or nearest to the value of the particular height 90, 92, 94,96, 98, or 100. If the height 90, 92, 94, 96, 98, or 100 is midwaybetween adjacent Cartesian coordinate values of Z in TABLE I, then thelesser or greater Cartesian coordinate value of Z may be selected forthe particular height 90, 92, 94, 96, 98, or 100. Alternatively, in someembodiments, if specific heights are desired but do not match thespecific Cartesian coordinate values of Z in TABLE I, then regressionanalysis and/or curve fitting may be used to analyze the data in TABLE Iand interpolate new Cartesian coordinate values of X, Y, Z to enable useof the desired heights. In some embodiments, if the Z values do notmatch the desired heights, then the portion 80 may be defined by theCartesian coordinate values of X, Y, and Z in TABLE I in the Z directionbetween the start and stop heights. Furthermore, with reference to TABLEI, the overall height dimension of each segment (e.g., 102, 104, 106)may be equal to an absolute value of the difference between the startand stop heights (e.g., Cartesian coordinate values of Z) for theparticular segment. In certain embodiments, an offset or correctionvalue may be used to account for negative values in TABLE I. Forexample, certain components of the airfoil 82, such as the bottom region70, may be disposed below the origin point 78 by a distance 108, andthus certain Z values may be negative denoting sections of airfoil 82below the origin point 78. Accordingly, the offset may be equal to anabsolute value of the smallest negative value given in TABLE I.

As mentioned earlier with respect to the portion 80 of FIG. 4, thesections 102, 104, and 106 (e.g., portion 80) of FIG. 5 may include oneor more airfoil shapes, profiles, or sections, such as an airfoil shapeshown in FIG. 6. More specifically, FIG. 6 is a cross-sectional top viewdepicting an embodiment of an airfoil shape 110 taken through line 6-6of FIGS. 3, 4, and 5, wherein the airfoil shape 110 may be included, forexample, in the portion 80 of the airfoil 82 (e.g., rotor blade 44,stator vane 50). As mentioned earlier, the airfoil 82 may be describedin terms of one or more airfoil shapes (e.g., sections) “stacked” on topof each other along the Z axis 74, such as the depicted airfoil shape110. To define the airfoil shape 110, a unique set or loci of points inspace are provided in TABLE I.

A Cartesian coordinate system of X, Y, and Z values given in TABLE Ibelow defines a suction side profile 112 and a pressure side profile 114of the airfoil shape 110 at various locations along the airfoil 82. Forexample, point 116 defines a first pair of suction side X and Y valuesat the Z value of line 6-6 of FIGS. 3-5, while point 118 defines asecond pair of pressure side X and Y values at the same Z value of FIGS.3-5. The X, Y, and Z values of TABLE I are set forth innon-dimensionalized units, and thus a variety of units of dimensions maybe used when the values are appropriately scaled by a scale factor F.The scale factor F may be substantially equal to 1, greater than 1, orless than 1. For example, the Cartesian coordinate values of X, Y, and Zmay be convertible to dimensional distances by multiplying the X, Y, andZ values by a constant number (e.g., 100). The number, used to convertthe non-dimensional values to dimensional distances, may be a fraction(e.g., ½, ¼, etc.), decimal fraction (e.g., 0.5, 1.5, 10.25, etc.),integer (e.g., 1, 2, 10, 100, etc.) or a mixed number (e.g., 1½, 10¼,etc.). The dimensional distances may be any suitable format (e.g.,inches, feet, millimeters, centimeters, etc.) These values exclude acoated region or coatings 120 of the airfoil 82. In other words, thesevalues correspond to the bare surface of the airfoil 82. The coatedregion 120 may include one or more coating layers, surface treatments,or a combination thereof, over the bare surface of the airfoil 82. TheCartesian coordinate system of FIG. 6 utilizes the sameorthogonally-related X axis 72, Y axis 76, and Z 74 as the Cartesiancoordinate system of FIGS. 3-5 to define locations on the airfoil shape110 for both the suction side profile 112 and the pressure side profile114 along the length L of the airfoil 82.

The X axis 72 and the Y axis 76 lie parallel to the base 60, asillustrated in the previous figures. In some embodiments, a positive Xcoordinate value is axial in the downstream axial direction 38 towardthe aft, for example the exhaust outlet 28 of the multi-stage axialcompressor 20. In some embodiments, a positive Y coordinate value isdirected normal to the X axis 72. A positive Z coordinate value isdirected radially from the X axis 72 and the Y axis 76 outward towardtip 68 of the airfoil 82, which is towards the static casing 48 of themulti-stage axial compressor 20 for rotor blades 44, and directedradially inward towards the shaft 30 of the multi-stage axial compressor20 for stator vanes 50.

By defining X and Y coordinate values at selected locations in a Zdirection normal to the X-Y plane, the airfoil shape 110, with itssuction side profile 112 and pressure side profile 114, may be defined,for example, by connecting each X and Y coordinate value to adjacent Xand Y coordinate values with smooth continuing arcs. The suction sideprofile 112 is joined to the pressure side profile 114, as shown in FIG.6, to define the airfoil shape 110. The airfoil shapes 110 (e.g.,sections) of the airfoil 82 at various surface locations (e.g., heights)between the Z coordinate values may be determined by smoothly connectingthe adjacent (e.g., “stacked”) airfoil shapes 110 to one another, thusforming the airfoil 82. It may be appreciated that the airfoil shape 110of the airfoil 82 may change from the base 60 to the tip 68. Forexample, adjacent airfoil shapes 110 may taper or expand in one or moredirections (e.g., X axis 72, Y axis 76), adjacent airfoil shapes 110 mayrotate about an axis (e.g., Z axis 74) in a clockwise direction or acounter-clockwise direction, or any combination thereof. It is also tobe noted that TABLE I values represent the Cartesian coordinate valuesfor the airfoil 82 at ambient, non-operating or non-hot conditions.Additionally, TABLE I values represent the Cartesian coordinate valuesfor an uncoated airfoil (i.e., without coatings 120). For a coatedairfoil, a thickness t of the coating 120 may be added to each of the X,Y values of TABLE I below.

The X, Y, and Z coordinate values of TABLE I below are non-dimensionalvalues convertible to dimensional distances with the scale factor F.That is, the X, Y, and Z values of TABLE I may be scaled as a functionof the same scale factor F (e.g., constant or number) to provide ascaled-up or a scaled-down airfoil. Thus, TABLE I defines therelationships between the X, Y, and Z coordinate values withoutspecifying the units of measure (e.g., dimensional units) for anembodiment of the airfoil 82. Accordingly, while different scale factorsF may be applied to the X, Y, and Z coordinate values of TABLE Itodefine different embodiments of the airfoil 82, each embodiment of theairfoil 82 regardless of the particular scale factor F is considered tobe defined by the X, Y, and Z coordinate values of TABLE I. For example,the X, Y, and Z coordinate values of TABLE I define a first embodimentof the airfoil 82 formed with a 1:1 inch scale factor F, a secondembodiment of the airfoil 82 formed with a 1:2 inch scale factor F, anda third embodiment of the airfoil 82 formed with a 1:1 cm scale factorF. It may be appreciated that any scale factor F may be used with the X,Y, and Z coordinate values of TABLE I, according to the designconsiderations of a particular embodiment.

The TABLE I values below are computer-generated and shown to fivedecimal places. However, certain values in TABLE I may be shown to lessthan five decimal places (e.g., 0, 1, 2, 3, or 4 decimal places),because the values are rounded to significant figures, the additionaldecimal places would merely show trailing zeroes, or a combinationthereof. Accordingly, in certain embodiments, any values having lessthan five decimal places may be shown with trailing zeroes out to 1, 2,3, 4, or 5 decimal places. Furthermore, in some embodiments and in viewof manufacturing constraints, actual values useful for forming theairfoil 82 are may be considered valid to fewer (e.g., one, two, three,or four) decimal places for determining the airfoil shape 110 of theairfoil 82. Further, there are typical manufacturing tolerances whichmay be accounted for in the airfoil shape 110. Accordingly, the X, Y,and Z values given in TABLE I are for the airfoil shape 110 of a nominalairfoil. It will therefore be appreciated that plus or minus typicalmanufacturing tolerances are applicable to these X, Y, and Z values andthat an airfoil 82 having a profile substantially in accordance withthose values includes such tolerances. For example, in certainembodiments, a manufacturing tolerance of about±between 0.001 to 0.20inches (e.g., between 0.025 to 5 mm) is within design limits for theairfoil 82, and a manufacturing tolerance of about±0.0008 to 0.1 inches(e.g., 0.02 to 2.5 mm) may be maintained during manufacturing.Accordingly, the values of X and Y carried to three decimal places andhaving a manufacturing tolerance about±0.010 inches (0.25 mm) andpreferably about±0.008 inches (0.20 mm) is acceptable to define theairfoil shape 110 of the airfoil 82 at each radial position (e.g., Zcoordinate, height) throughout its entire length. As used herein, anyreference to Cartesian coordinate values of X, Y, and Z as set forth inTABLE I to a maximum of N decimal places is intended to include: (1)values to N decimal places if shown in TABLE I with N or greater decimalplaces, and (2) values to less than N decimal places if shown in TABLE 1with less than N decimal places, wherein N may be 0, 1, 2, 3, 4, or 5.For example, any reference to Cartesian coordinate values of X, Y, and Zas set forth in TABLE I to a maximum of three (3) decimal places isintended to include: (1) values to three (3) decimal places if shown inTABLE I with three (3) or greater decimal places, and (2) values to lessthan three (3) decimal places if shown in TABLE 1 with less than three(3) decimal places (e.g., 0, 1, or 2 decimal places). Furthermore, anyreference to Cartesian coordinate values of X, Y, and Z as set forth inTABLE I all carried to N decimal places is intended to include: (1)values to N decimal places if shown in TABLE I with N or greater decimalplaces, and (2) values with trailing zeros to N decimal places if shownin TABLE 1 with less than N decimal places, wherein N may be 0, 1, 2, 3,4, or 5.

As noted previously, the airfoil 82 may also be coated for protectionagainst corrosion, erosion, wear, and oxidation after the airfoil 82 ismanufactured, according to the values of TABLE I and within thetolerances explained above. For example, the coating region 120 mayinclude one or more corrosion resistant layers, erosion resistantlayers, wear resistant layers, oxidation resistant or anti-oxidationlayers, or any combination thereof. An anti-corrosion coating may beprovided with an average thickness t of 0.008 inches (0.20 mm), between0.001 and 0.1 inches (between 0.25 and 2.5 mm), between, 0.0001 and 1inches or more (between 0.0025 and 2.5 mm or more). For example, incertain embodiments, the coating 120 may increase X and Y values of asuction side in TABLE I by no greater than approximately 3.5 mm along afirst suction portion, a first pressure portion, or both. It is to benoted that additional anti-oxidation coatings 120 may be provided, suchas overcoats.

TABLE I Pressure Side or Surface Suction Side or Surface X Y Z X Y Z−1.4465 2.02672 −0.2253 2.01162 −1.7412 −0.2253 −1.4454 2.02712 −0.22532.00618 −1.7472 −0.2253 −1.4435 2.02778 −0.2253 1.99717 −1.7528 −0.2253−1.4395 2.02871 −0.2253 1.98406 −1.7556 −0.2253 −1.4311 2.0295 −0.22531.96855 −1.7523 −0.2253 −1.4182 2.02818 −0.2253 1.95 −1.7413 −0.2253−1.3962 2.02036 −0.2253 1.92602 −1.7269 −0.2253 −1.3697 2.00406 −0.22531.89608 −1.7089 −0.2253 −1.3383 1.97743 −0.2253 1.85977 −1.687 −0.2253−1.3027 1.9402 −0.2253 1.81644 −1.6609 −0.2253 −1.2593 1.88905 −0.22531.76596 −1.6303 −0.2253 −1.2104 1.82903 −0.2253 1.7062 −1.594 −0.2253−1.1598 1.76384 −0.2253 1.63744 −1.5518 −0.2253 −1.1037 1.68991 −0.22531.55953 −1.504 −0.2253 −1.0423 1.60696 −0.2253 1.47274 −1.4502 −0.2253−0.9753 1.5154 −0.2253 1.37707 −1.3902 −0.2253 −0.9055 1.41934 −0.22531.27266 −1.3241 −0.2253 −0.8329 1.31917 −0.2253 1.16415 −1.2543 −0.2253−0.7567 1.21503 −0.2253 1.05179 −1.1806 −0.2253 −0.6771 1.1073 −0.22530.93558 −1.1031 −0.2253 −0.5935 0.99587 −0.2253 0.81567 −1.0213 −0.2253−0.5056 0.88126 −0.2253 0.69245 −0.9352 −0.2253 −0.4137 0.76347 −0.22530.56604 −0.8442 −0.2253 −0.3172 0.64236 −0.2253 0.43659 −0.7482 −0.2253−0.2196 0.52232 −0.2253 0.30886 −0.6499 −0.2253 −0.1208 0.4032 −0.22530.18272 −0.5492 −0.2253 −0.0212 0.28488 −0.2253 0.05883 −0.446 −0.22530.07963 0.16748 −0.2253 −0.0627 −0.3404 −0.2253 0.18166 0.05128 −0.2253−0.1817 −0.2319 −0.2253 0.28488 −0.064 −0.2253 −0.298 −0.1206 −0.22530.38942 −0.178 −0.2253 −0.4114 −0.0062 −0.2253 0.49529 −0.2908 −0.2253−0.5217 0.11143 −0.2253 0.60261 −0.4025 −0.2253 −0.6282 0.23241 −0.22530.711 −0.5128 −0.2253 −0.7305 0.35629 −0.2253 0.82071 −0.622 −0.2253−0.8281 0.48296 −0.2253 0.9279 −0.7264 −0.2253 −0.9181 0.60818 −0.22531.03231 −0.826 −0.2253 −1.0009 0.73167 −0.2253 1.1338 −0.9214 −0.2253−1.0758 0.8537 −0.2253 1.23225 −1.0124 −0.2253 −1.1437 0.97374 −0.22531.32765 −1.0992 −0.2253 −1.2067 1.09087 −0.2253 1.41974 −1.182 −0.2253−1.2617 1.20668 −0.2253 1.50838 −1.261 −0.2253 −1.3103 1.3201 −0.22531.5896 −1.3327 −0.2253 −1.3543 1.43021 −0.2253 1.66341 −1.3972 −0.2253−1.3907 1.53197 −0.2253 1.72952 −1.4549 −0.2253 −1.4199 1.62485 −0.22531.78796 −1.5055 −0.2253 −1.4432 1.70819 −0.2253 1.8387 −1.5495 −0.2253−1.4619 1.78703 −0.2253 1.88163 −1.5866 −0.2253 −1.4754 1.85606 −0.22531.91834 −1.6181 −0.2253 −1.4824 1.90959 −0.2253 1.94934 −1.6447 −0.2253−1.4829 1.95279 −0.2253 1.97478 −1.6665 −0.2253 −1.4778 1.98485 −0.22531.99519 −1.6839 −0.2253 −1.4685 2.00724 −0.2253 2.01056 −1.6977 −0.2253−1.4602 2.01784 −0.2253 2.01731 −1.7115 −0.2253 −1.4533 2.02328 −0.22532.01811 −1.7244 −0.2253 −1.4496 2.0254 −0.2253 2.01533 −1.7344 −0.2253−1.4476 2.02632 −0.2253 −1.468 1.95146 0 1.96802 −1.736 0 −1.467 1.951860 1.96259 −1.7416 0 −1.4651 1.95265 0 1.95358 −1.7469 0 −1.4612 1.953850 1.94073 −1.7491 0 −1.4531 1.9553 0 1.92562 −1.7452 0 −1.4403 1.95491 01.90774 −1.7342 0 −1.4182 1.94841 0 1.88455 −1.7197 0 −1.3913 1.93371 01.85553 −1.7017 0 −1.3593 1.90893 0 1.82029 −1.6797 0 −1.3232 1.87342 01.77842 −1.6536 0 −1.2786 1.82453 0 1.72939 −1.623 0 −1.2289 1.76676 01.67149 −1.5867 0 −1.1773 1.70382 0 1.60471 −1.5448 0 −1.1204 1.63227 01.52918 −1.497 0 −1.0582 1.55197 0 1.44505 −1.4435 0 −0.9903 1.46307 01.35216 −1.3838 0 −0.9198 1.36992 0 1.2508 −1.318 0 −0.8462 1.27253 01.14546 −1.2487 0 −0.7692 1.17157 0 1.03628 −1.1755 0 −0.6885 1.06702 00.92339 −1.0987 0 −0.6039 0.95904 0 0.80719 −1.0176 0 −0.5152 0.848 00.68754 −0.9323 0 −0.4222 0.73365 0 0.56472 −0.8423 0 −0.3249 0.61639 00.43924 −0.7474 0 −0.2267 0.50006 0 0.31522 −0.6504 0 −0.1276 0.38425 00.19279 −0.5512 0 −0.0277 0.26924 0 0.07221 −0.4497 0 0.07327 0.15489 0−0.0465 −0.3457 0 0.17503 0.04134 0 −0.1628 −0.2393 0 0.27772 −0.0713 0−0.2767 −0.1304 0 0.38147 −0.1831 0 −0.388 −0.019 0 0.48614 −0.2939 0−0.4965 0.0954 0 0.59175 −0.4037 0 −0.6021 0.21266 0 0.69841 −0.5126 0−0.7042 0.33284 0 0.80613 −0.6205 0 −0.8028 0.45567 0 0.91107 −0.724 0−0.8945 0.57717 0 1.01323 −0.8231 0 −0.9796 0.69708 0 1.11247 −0.9178 0−1.0576 0.81567 0 1.20853 −1.0086 0 −1.129 0.93267 0 1.30155 −1.0953 0−1.1953 1.04635 0 1.39125 −1.1779 0 −1.2539 1.15832 0 1.47751 −1.2569 0−1.3062 1.26789 0 1.55661 −1.3286 0 −1.3536 1.37363 0 1.62843 −1.3931 0−1.3935 1.47128 0 1.69282 −1.4509 0 −1.4258 1.56045 0 1.74966 −1.5016 0−1.4521 1.64062 0 1.79909 −1.5455 0 −1.4737 1.7168 0 1.84082 −1.5827 0−1.4896 1.78345 0 1.8766 −1.6144 0 −1.4984 1.83539 0 1.90668 −1.641 0−1.5004 1.87739 0 1.93145 −1.6629 0 −1.4966 1.90893 0 1.95133 −1.6804 0−1.4884 1.93106 0 1.96643 −1.6939 0 −1.4807 1.94192 0 1.97372 −1.7071 0−1.4743 1.94762 0 1.97465 −1.7197 0 −1.4708 1.94987 0 1.97187 −1.7295 0−1.4689 1.95093 0 −1.4945 1.85924 0.3233 1.90363 −1.7335 0.3233 −1.49351.85964 0.3233 1.8982 −1.7388 0.3233 −1.4917 1.86057 0.3233 1.88932−1.7437 0.3233 −1.4878 1.86189 0.3233 1.87673 −1.7456 0.3233 −1.481.86361 0.3233 1.86216 −1.7413 0.3233 −1.4674 1.86348 0.3233 1.84506−1.7299 0.3233 −1.4457 1.85765 0.3233 1.82294 −1.7152 0.3233 −1.41891.844 0.3233 1.79524 −1.6968 0.3233 −1.3869 1.82068 0.3233 1.76146−1.6745 0.3233 −1.3502 1.78703 0.3233 1.72144 −1.6479 0.3233 −1.30551.74012 0.3233 1.67454 −1.6168 0.3233 −1.2554 1.68474 0.3233 1.61902−1.5799 0.3233 −1.2031 1.62458 0.3233 1.55515 −1.5374 0.3233 −1.14491.55635 0.3233 1.48268 −1.489 0.3233 −1.0812 1.47976 0.3233 1.40198−1.4348 0.3233 −1.0115 1.39523 0.3233 1.31281 −1.3748 0.3233 −0.93891.30645 0.3233 1.21542 −1.3087 0.3233 −0.8634 1.21383 0.3233 1.11406−1.2391 0.3233 −0.7845 1.11737 0.3233 1.00912 −1.1661 0.3233 −0.70231.01734 0.3233 0.9006 −1.0894 0.3233 −0.6165 0.91372 0.3233 0.78864−1.0089 0.3233 −0.5271 0.80666 0.3233 0.67363 −0.9242 0.3233 −0.43370.69642 0.3233 0.55571 −0.8352 0.3233 −0.3363 0.58313 0.3233 0.435−0.7416 0.3233 −0.2381 0.47051 0.3233 0.31575 −0.6465 0.3233 −0.13910.35841 0.3233 0.19809 −0.5495 0.3233 −0.0398 0.24685 0.3233 0.08215−0.4506 0.3233 0.06042 0.13581 0.3233 −0.0319 −0.3498 0.3233 0.161250.02544 0.3233 −0.1442 −0.2469 0.3233 0.26275 −0.0843 0.3233 −0.2543−0.1416 0.3233 0.36504 −0.1933 0.3233 −0.3624 −0.0339 0.3233 0.46786−0.3017 0.3233 −0.468 0.07632 0.3233 0.57147 −0.4094 0.3233 −0.57120.18934 0.3233 0.67588 −0.5165 0.3233 −0.6719 0.30502 0.3233 0.78096−0.6228 0.3233 −0.77 0.42374 0.3233 0.88325 −0.7248 0.3233 −0.86230.54113 0.3233 0.98262 −0.8227 0.3233 −0.949 0.65694 0.3233 1.07895−0.9166 0.3233 −1.0293 0.77075 0.3233 1.17223 −1.0066 0.3233 −1.10360.88245 0.3233 1.26233 −1.0927 0.3233 −1.173 0.99123 0.3233 1.34912−1.175 0.3233 −1.2361 1.09776 0.3233 1.43259 −1.2536 0.3233 −1.29311.20164 0.3233 1.50891 −1.3251 0.3233 −1.3455 1.30234 0.3233 1.57808−1.3898 0.3233 −1.3902 1.39536 0.3233 1.63995 −1.4477 0.3233 −1.42771.48042 0.3233 1.69468 −1.4986 0.3233 −1.4588 1.55688 0.3233 1.74211−1.5428 0.3233 −1.4851 1.62975 0.3233 1.78226 −1.5802 0.3233 −1.50531.69375 0.3233 1.81658 −1.612 0.3233 −1.517 1.7441 0.3233 1.84546−1.6388 0.3233 −1.5215 1.78491 0.3233 1.86918 −1.6609 0.3233 −1.51951.81591 0.3233 1.88813 −1.6784 0.3233 −1.513 1.83817 0.3233 1.9027−1.692 0.3233 −1.5063 1.84917 0.3233 1.90972 −1.7051 0.3233 −1.50041.85513 0.3233 1.91039 −1.7175 0.3233 −1.4971 1.85752 0.3233 1.9076−1.727 0.3233 −1.4954 1.85871 0.3233 −1.4831 1.82506 0.53 1.84917−1.7468 0.53 −1.4823 1.82545 0.53 1.84374 −1.7519 0.53 −1.4804 1.826380.53 1.83486 −1.7566 0.53 −1.4767 1.82784 0.53 1.82254 −1.7581 0.53−1.4689 1.82943 0.53 1.80823 −1.7535 0.53 −1.4564 1.82916 0.53 1.7918−1.742 0.53 −1.435 1.82333 0.53 1.77033 −1.7269 0.53 −1.4087 1.809690.53 1.74344 −1.7081 0.53 −1.3772 1.7865 0.53 1.71084 −1.6853 0.53−1.341 1.75311 0.53 1.67202 −1.6581 0.53 −1.2972 1.70647 0.53 1.62657−1.6263 0.53 −1.2482 1.65122 0.53 1.57291 −1.5888 0.53 −1.1969 1.591460.53 1.5109 −1.5455 0.53 −1.1398 1.52375 0.53 1.44067 −1.4963 0.53−1.0768 1.44809 0.53 1.36223 −1.4413 0.53 −1.0081 1.36435 0.53 1.27571−1.3804 0.53 −0.9364 1.27677 0.53 1.18111 −1.3135 0.53 −0.8618 1.185210.53 1.08279 −1.2431 0.53 −0.784 1.08995 0.53 0.98063 −1.1693 0.53−0.7028 0.99097 0.53 0.87516 −1.0919 0.53 −0.6183 0.88855 0.53 0.76625−1.0107 0.53 −0.5301 0.78268 0.53 0.65442 −0.9255 0.53 −0.4382 0.67350.53 0.53954 −0.8359 0.53 −0.3423 0.56127 0.53 0.42201 −0.742 0.53−0.2455 0.44984 0.53 0.30594 −0.6466 0.53 −0.1481 0.33894 0.53 0.19133−0.5494 0.53 −0.0502 0.22843 0.53 0.07844 −0.4505 0.53 0.04823 0.118590.53 −0.0329 −0.3497 0.53 0.14747 0.00928 0.53 −0.1422 −0.2469 0.530.24725 −0.0996 0.53 −0.2496 −0.1419 0.53 0.34768 −0.2078 0.53 −0.355−0.0347 0.53 0.44878 −0.3155 0.53 −0.4581 0.075 0.53 0.55041 −0.42240.53 −0.5588 0.18722 0.53 0.6527 −0.5288 0.53 −0.6572 0.30197 0.530.75565 −0.6347 0.53 −0.7533 0.41936 0.53 0.85582 −0.7363 0.53 −0.84380.53517 0.53 0.95307 −0.834 0.53 −0.929 0.64899 0.53 1.04728 −0.92760.53 −1.0082 0.76068 0.53 1.13844 −1.0175 0.53 −1.082 0.87013 0.531.22629 −1.1036 0.53 −1.151 0.97666 0.53 1.31096 −1.186 0.53 −1.21441.0808 0.53 1.39218 −1.2649 0.53 −1.2721 1.1823 0.53 1.46651 −1.33670.53 −1.3253 1.28048 0.53 1.53369 −1.4016 0.53 −1.371 1.37124 0.531.59398 −1.4598 0.53 −1.4095 1.45406 0.53 1.64698 −1.511 0.53 −1.44171.52865 0.53 1.69309 −1.5556 0.53 −1.4694 1.59954 0.53 1.73191 −1.59320.53 −1.4908 1.66208 0.53 1.76517 −1.6253 0.53 −1.5031 1.71124 0.531.79312 −1.6523 0.53 −1.5081 1.75152 0.53 1.81618 −1.6745 0.53 −1.50681.78199 0.53 1.83473 −1.6922 0.53 −1.501 1.80399 0.53 1.84877 −1.70590.53 −1.4946 1.81499 0.53 1.85566 −1.7189 0.53 −1.489 1.82095 0.531.85619 −1.7313 0.53 −1.4857 1.82333 0.53 1.85315 −1.7407 0.53 −1.4841.82453 0.53 −1.4355 1.79312 0.86589 1.7445 −1.7872 0.86589 −1.43461.79352 0.86589 1.73893 −1.7919 0.86589 −1.4329 1.79445 0.86589 1.73019−1.7962 0.86589 −1.4292 1.79577 0.86589 1.718 −1.7972 0.86589 −1.42151.7971 0.86589 1.70435 −1.7921 0.86589 −1.4093 1.79644 0.86589 1.68858−1.7803 0.86589 −1.3885 1.78968 0.86589 1.66831 −1.7648 0.86589 −1.36321.77537 0.86589 1.64274 −1.7454 0.86589 −1.3328 1.75152 0.86589 1.61186−1.722 0.86589 −1.2982 1.71773 0.86589 1.57503 −1.694 0.86589 −1.25661.67083 0.86589 1.53197 −1.6613 0.86589 −1.2107 1.61518 0.86589 1.48095−1.6225 0.86589 −1.1622 1.55502 0.86589 1.42226 −1.5778 0.86589 −1.10821.48705 0.86589 1.35574 −1.5271 0.86589 −1.0487 1.41126 0.86589 1.28154−1.4702 0.86589 −0.9833 1.32778 0.86589 1.19966 −1.4073 0.86589 −0.91521.24033 0.86589 1.11022 −1.338 0.86589 −0.844 1.14904 0.86589 1.0172−1.2652 0.86589 −0.7698 1.05404 0.86589 0.92061 −1.1892 0.86589 −0.69250.95546 0.86589 0.82084 −1.1094 0.86589 −0.6118 0.8533 0.86589 0.71775−1.026 0.86589 −0.5275 0.7477 0.86589 0.61175 −0.9385 0.86589 −0.43960.63905 0.86589 0.50297 −0.8469 0.86589 −0.3478 0.52722 0.86589 0.39154−0.751 0.86589 −0.2552 0.41618 0.86589 0.28143 −0.6538 0.86589 −0.16170.30581 0.86589 0.17278 −0.555 0.86589 −0.0674 0.19584 0.86589 0.06559−0.4547 0.86589 0.02716 0.08639 0.86589 −0.04 −0.3529 0.86589 0.1223−0.0225 0.86589 −0.1438 −0.2491 0.86589 0.21796 −0.131 0.86589 −0.2458−0.1435 0.86589 0.31416 −0.239 0.86589 −0.3457 −0.0356 0.86589 0.41075−0.3466 0.86589 −0.4434 0.07433 0.86589 0.50801 −0.4537 0.86589 −0.53890.18656 0.86589 0.60579 −0.5603 0.86589 −0.6322 0.30104 0.86589 0.70411−0.6665 0.86589 −0.7235 0.41777 0.86589 0.79951 −0.7688 0.86589 −0.80960.53252 0.86589 0.89212 −0.8671 0.86589 −0.8904 0.64501 0.86589 0.98183−0.9616 0.86589 −0.9662 0.75512 0.86589 1.06848 −1.0522 0.86589 −1.03680.86271 0.86589 1.15196 −1.1391 0.86589 −1.1031 0.96725 0.86589 1.23238−1.2225 0.86589 −1.1644 1.06901 0.86589 1.3095 −1.3021 0.86589 −1.22071.16799 0.86589 1.38012 −1.3747 0.86589 −1.2727 1.26379 0.86589 1.44412−1.4403 0.86589 −1.3177 1.3519 0.86589 1.50149 −1.4988 0.86589 −1.35631.43233 0.86589 1.55211 −1.5507 0.86589 −1.3889 1.50454 0.86589 1.59596−1.5954 0.86589 −1.4172 1.57317 0.86589 1.63306 −1.6333 0.86589 −1.43911.63373 0.86589 1.66473 −1.6657 0.86589 −1.4519 1.68156 0.86589 1.6915−1.6928 0.86589 −1.4576 1.72078 0.86589 1.71349 −1.7152 0.86589 −1.45721.75046 0.86589 1.73111 −1.733 0.86589 −1.4522 1.77219 0.86589 1.74476−1.7466 0.86589 −1.4465 1.78305 0.86589 1.75139 −1.7595 0.86589 −1.44121.78902 0.86589 1.75178 −1.7718 0.86589 −1.438 1.79153 0.86589 1.74874−1.7811 0.86589 −1.4364 1.79259 0.86589 −1.3868 1.77351 1.14069 1.65347−1.8251 1.14069 −1.386 1.77391 1.14069 1.6479 −1.8297 1.14069 −1.38421.77484 1.14069 1.63916 −1.8335 1.14069 −1.3805 1.77603 1.14069 1.62723−1.8343 1.14069 −1.373 1.77722 1.14069 1.61398 −1.8289 1.14069 −1.36091.77603 1.14069 1.59888 −1.8168 1.14069 −1.3406 1.76874 1.14069 1.5794−1.8011 1.14069 −1.3163 1.7539 1.14069 1.55502 −1.7812 1.14069 −1.2871.72966 1.14069 1.52547 −1.7572 1.14069 −1.2539 1.69547 1.14069 1.49023−1.7286 1.14069 −1.2142 1.64817 1.14069 1.44902 −1.6951 1.14069 −1.17061.59199 1.14069 1.40039 −1.6555 1.14069 −1.1248 1.53157 1.14069 1.34435−1.6096 1.14069 −1.0737 1.4632 1.14069 1.28088 −1.5575 1.14069 −1.01711.38701 1.14069 1.21012 −1.4991 1.14069 −0.9549 1.30314 1.14069 1.13208−1.4345 1.14069 −0.8901 1.21542 1.14069 1.04675 −1.3633 1.14069 −0.82231.12387 1.14069 0.95824 −1.2886 1.14069 −0.7515 1.02873 1.14069 0.86629−1.2104 1.14069 −0.6777 0.92989 1.14069 0.77128 −1.1285 1.14069 −0.60060.8276 1.14069 0.67323 −1.043 1.14069 −0.5201 0.72199 1.14069 0.57227−0.9535 1.14069 −0.4359 0.61321 1.14069 0.46865 −0.8599 1.14069 −0.34810.50125 1.14069 0.36252 −0.7621 1.14069 −0.2592 0.39008 1.14069 0.25758−0.6632 1.14069 −0.1693 0.27958 1.14069 0.15383 −0.5629 1.14069 −0.0790.16973 1.14069 0.05168 −0.4611 1.14069 0.01206 0.06016 1.14069 −0.0489−0.3579 1.14069 0.10335 −0.0489 1.14069 −0.1479 −0.2531 1.14069 0.19531−0.1577 1.14069 −0.245 −0.1464 1.14069 0.28753 −0.2659 1.14069 −0.3401−0.0379 1.14069 0.38028 −0.3739 1.14069 −0.433 0.07274 1.14069 0.47342−0.4815 1.14069 −0.5239 0.1855 1.14069 0.56697 −0.5888 1.14069 −0.6130.30011 1.14069 0.66091 −0.6958 1.14069 −0.6999 0.41658 1.14069 0.7522−0.7987 1.14069 −0.7819 0.53093 1.14069 0.84058 −0.898 1.14069 −0.8590.64276 1.14069 0.92618 −0.9932 1.14069 −0.9313 0.75207 1.14069 1.00872−1.0849 1.14069 −0.9989 0.8586 1.14069 1.08849 −1.1728 1.14069 −1.06230.96208 1.14069 1.16507 −1.2569 1.14069 −1.1212 1.06265 1.14069 1.23888−1.3375 1.14069 −1.1755 1.16017 1.14069 1.30619 −1.4107 1.14069 −1.22581.25451 1.14069 1.3674 −1.4771 1.14069 −1.2695 1.34117 1.14069 1.42212−1.5362 1.14069 −1.3073 1.42 1.14069 1.47062 −1.5884 1.14069 −1.33921.49076 1.14069 1.51262 −1.6336 1.14069 −1.3675 1.55794 1.14069 1.54813−1.6718 1.14069 −1.3891 1.61716 1.14069 1.57861 −1.7044 1.14069 −1.40191.66407 1.14069 1.60418 −1.7318 1.14069 −1.4076 1.70236 1.14069 1.62538−1.7542 1.14069 −1.4076 1.73151 1.14069 1.64221 −1.7722 1.14069 −1.40291.75284 1.14069 1.65519 −1.7861 1.14069 −1.3975 1.76358 1.14069 1.66102−1.7988 1.14069 −1.3924 1.76954 1.14069 1.66102 −1.8105 1.14069 −1.38931.77192 1.14069 1.65771 −1.8194 1.14069 −1.3877 1.77298 1.14069 −1.31731.74821 1.52017 1.53422 −1.8694 1.52017 −1.3164 1.7486 1.52017 1.52879−1.8738 1.52017 −1.3147 1.7494 1.52017 1.52004 −1.8774 1.52017 −1.31111.75059 1.52017 1.50838 −1.8778 1.52017 −1.3035 1.75139 1.52017 1.49553−1.8722 1.52017 −1.2919 1.74966 1.52017 1.48135 −1.8599 1.52017 −1.27241.74158 1.52017 1.4628 −1.8437 1.52017 −1.2492 1.72608 1.52017 1.43975−1.8237 1.52017 −1.2213 1.7013 1.52017 1.41166 −1.7992 1.52017 −1.18991.66685 1.52017 1.37827 −1.7699 1.52017 −1.1528 1.61955 1.52017 1.33931−1.7358 1.52017 −1.1122 1.56297 1.52017 1.29333 −1.6952 1.52017 −1.06971.50215 1.52017 1.24047 −1.6484 1.52017 −1.0222 1.43352 1.52017 1.18058−1.5952 1.52017 −0.9698 1.35707 1.52017 1.1138 −1.5354 1.52017 −0.9121.27293 1.52017 1.04026 −1.469 1.52017 −0.8517 1.18495 1.52017 0.9601−1.3962 1.52017 −0.7886 1.09326 1.52017 0.87675 −1.3196 1.52017 −0.72270.99786 1.52017 0.79036 −1.2395 1.52017 −0.6536 0.89901 1.52017 0.70119−1.1557 1.52017 −0.5815 0.79659 1.52017 0.6091 −1.068 1.52017 −0.50620.69099 1.52017 0.5145 −0.9764 1.52017 −0.4272 0.58194 1.52017 0.41738−0.8807 1.52017 −0.3448 0.46985 1.52017 0.31787 −0.7808 1.52017 −0.26140.35855 1.52017 0.21969 −0.6799 1.52017 −0.1772 0.24778 1.52017 0.1227−0.5777 1.52017 −0.0921 0.1378 1.52017 0.02716 −0.4744 1.52017 −0.00640.02822 1.52017 −0.0669 −0.3697 1.52017 0.07963 −0.0811 1.52017 −0.1593−0.2634 1.52017 0.16616 −0.1901 1.52017 −0.2498 −0.1556 1.52017 0.25294−0.2989 1.52017 −0.3383 −0.046 1.52017 0.34 −0.4073 1.52017 −0.42480.06546 1.52017 0.42758 −0.5154 1.52017 −0.5095 0.17848 1.52017 0.51543−0.6234 1.52017 −0.5924 0.29309 1.52017 0.60354 −0.731 1.52017 −0.67340.40943 1.52017 0.689 −0.8349 1.52017 −0.75 0.52338 1.52017 0.77181−0.9349 1.52017 −0.8219 0.63468 1.52017 0.85184 −1.0311 1.52017 −0.88950.74319 1.52017 0.92922 −1.1236 1.52017 −0.9528 0.8488 1.52017 1.00382−1.2124 1.52017 −1.012 0.95135 1.52017 1.07577 −1.2973 1.52017 −1.0671.05073 1.52017 1.1448 −1.3785 1.52017 −1.118 1.14692 1.52017 1.20814−1.4526 1.52017 −1.1653 1.2398 1.52017 1.26551 −1.5194 1.52017 −1.20671.325 1.52017 1.31705 −1.5791 1.52017 −1.2426 1.40238 1.52017 1.36263−1.6317 1.52017 −1.2731 1.47181 1.52017 1.40212 −1.6772 1.52017 −1.30011.5374 1.52017 1.43564 −1.7156 1.52017 −1.3205 1.59557 1.52017 1.46439−1.7485 1.52017 −1.3322 1.64141 1.52017 1.48851 −1.776 1.52017 −1.33751.67891 1.52017 1.50838 −1.7987 1.52017 −1.3373 1.70726 1.52017 1.52428−1.8168 1.52017 −1.333 1.72807 1.52017 1.53647 −1.8308 1.52017 −1.32781.73867 1.52017 1.5423 −1.8433 1.52017 −1.3228 1.74436 1.52017 1.54204−1.8551 1.52017 −1.3197 1.74675 1.52017 1.53859 −1.8639 1.52017 −1.31811.74781 1.52017 −1.2368 1.7009 2.07084 1.40172 −1.8936 2.07084 −1.23581.7013 2.07084 1.39629 −1.8975 2.07084 −1.2341 1.70196 2.07084 1.38781−1.901 2.07084 −1.2307 1.70276 2.07084 1.37641 −1.9012 2.07084 −1.22321.70316 2.07084 1.36396 −1.8959 2.07084 −1.2121 1.70077 2.07084 1.35031−1.8838 2.07084 −1.1938 1.69189 2.07084 1.33269 −1.8679 2.07084 −1.17211.67573 2.07084 1.31069 −1.848 2.07084 −1.146 1.65069 2.07084 1.28393−1.8237 2.07084 −1.1167 1.6165 2.07084 1.25213 −1.7949 2.07084 −1.08211.5696 2.07084 1.21503 −1.7609 2.07084 −1.0448 1.51368 2.07084 1.17117−1.7208 2.07084 −1.0061 1.45313 2.07084 1.12082 −1.6743 2.07084 −0.9631.38489 2.07084 1.06398 −1.6214 2.07084 −0.9152 1.30884 2.07084 1.00064−1.5619 2.07084 −0.8627 1.22523 2.07084 0.93108 −1.4959 2.07084 −0.80761.13791 2.07084 0.85516 −1.4232 2.07084 −0.75 1.04688 2.07084 0.77658−1.3467 2.07084 −0.6895 0.95228 2.07084 0.69523 −1.2666 2.07084 −0.62630.85423 2.07084 0.61149 −1.1824 2.07084 −0.5601 0.75273 2.07084 0.52536−1.0946 2.07084 −0.4908 0.64793 2.07084 0.43699 −1.0025 2.07084 −0.41820.53981 2.07084 0.34662 −0.9063 2.07084 −0.3421 0.42864 2.07084 0.25427−0.8059 2.07084 −0.2651 0.31813 2.07084 0.16337 −0.7042 2.07084 −0.18740.20829 2.07084 0.07394 −0.6016 2.07084 −0.1085 0.09911 2.07084 −0.0141−0.4975 2.07084 −0.029 −0.0095 2.07084 −0.1004 −0.3923 2.07084 0.05088−0.1179 2.07084 −0.185 −0.2857 2.07084 0.13131 −0.2259 2.07084 −0.2678−0.1776 2.07084 0.21187 −0.3338 2.07084 −0.3487 −0.068 2.07084 0.29296−0.4414 2.07084 −0.4278 0.04306 2.07084 0.37431 −0.5487 2.07084 −0.50540.15542 2.07084 0.45593 −0.6557 2.07084 −0.5812 0.26924 2.07084 0.53782−0.7627 2.07084 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2.07084 −1.1938 1.62591 2.639 1.31294−1.882 2.639 −1.193 1.62631 2.639 1.30778 −1.886 2.639 −1.1913 1.626842.639 1.29969 −1.8895 2.639 −1.1879 1.62763 2.639 1.2887 −1.8901 2.639−1.1808 1.62763 2.639 1.27637 −1.8857 2.639 −1.1701 1.62498 2.6391.26273 −1.8745 2.639 −1.1528 1.61557 2.639 1.24524 −1.8594 2.639−1.1322 1.59941 2.639 1.22337 −1.8406 2.639 −1.1077 1.5745 2.639 1.19701−1.8176 2.639 −1.08 1.54084 2.639 1.1656 −1.7902 2.639 −1.0473 1.4952.639 1.12903 −1.758 2.639 −1.012 1.44028 2.639 1.08597 −1.7197 2.639−0.9759 1.38105 2.639 1.03655 −1.6752 2.639 −0.9356 1.31414 2.639 0.9809−1.6245 2.639 −0.8909 1.23967 2.639 0.91915 −1.5672 2.639 −0.84181.15779 2.639 0.85145 −1.5035 2.639 −0.7901 1.07232 2.639 0.77791 −1.4332.639 −0.736 0.98328 2.639 0.70185 −1.3585 2.639 −0.6792 0.89093 2.6390.62341 −1.2804 2.639 −0.6197 0.795 2.639 0.54299 −1.1981 2.639 −0.55720.69589 2.639 0.46057 −1.1119 2.639 −0.4917 0.5936 2.639 0.3763 −1.02162.639 −0.4229 0.488 2.639 0.29031 −0.927 2.639 −0.3509 0.37948 2.6390.20286 −0.8281 2.639 −0.278 0.27163 2.639 0.11687 −0.728 2.639 −0.20410.16443 2.639 0.03246 −0.6267 2.639 −0.1292 0.0579 2.639 −0.0505 −0.52432.639 −0.0534 −0.048 2.639 −0.1317 −0.4206 2.639 0.02306 −0.1533 2.639−0.2113 −0.3156 2.639 0.09991 −0.2584 2.639 −0.2891 −0.2094 2.6390.17715 −0.3632 2.639 −0.3654 −0.1019 2.639 0.2548 −0.4679 2.639 −0.440.00689 2.639 0.33271 −0.5721 2.639 −0.5132 0.11673 2.639 0.41088−0.6763 2.639 −0.5846 0.22803 2.639 0.48946 −0.7802 2.639 −0.65420.34053 2.639 0.56564 −0.8803 2.639 −0.7197 0.45077 2.639 0.63931−0.9769 2.639 −0.7814 0.55836 2.639 0.71073 −1.0698 2.639 −0.8391 0.66332.639 0.77963 −1.1592 2.639 −0.8932 0.76532 2.639 0.84601 −1.2451 2.639−0.9435 0.8643 2.639 0.90988 −1.3274 2.639 −0.9902 0.9601 2.639 0.97136−1.4061 2.639 −1.0336 1.05258 2.639 1.02754 −1.4779 2.639 −1.07351.14175 2.639 1.07842 −1.5427 2.639 −1.1085 1.22337 2.639 1.124 −1.60072.639 −1.1387 1.29744 2.639 1.16441 −1.6519 2.639 −1.1643 1.36369 2.6391.19939 −1.6961 2.639 −1.186 1.42663 2.639 1.22907 −1.7335 2.639 −1.20151.48228 2.639 1.25438 −1.7654 2.639 −1.2101 1.52587 2.639 1.27584−1.7923 2.639 −1.2137 1.56125 2.639 1.29333 −1.8143 2.639 −1.21291.58775 2.639 1.30738 −1.8321 2.639 −1.2088 1.60736 2.639 1.31798−1.8457 2.639 −1.2038 1.6173 2.639 1.32195 −1.8582 2.639 −1.1991 1.62262.639 1.32076 −1.8691 2.639 −1.1962 1.62472 2.639 1.31718 −1.8769 2.639−1.1946 1.62551 2.639 −1.1856 1.55409 3.03796 1.27916 −1.8614 3.03796−1.1848 1.55449 3.03796 1.27399 −1.8653 3.03796 −1.1832 1.55502 3.037961.26617 −1.8689 3.03796 −1.1798 1.55582 3.03796 1.25544 −1.8698 3.03796−1.1729 1.55568 3.03796 1.24325 −1.8659 3.03796 −1.1626 1.55277 3.037961.22947 −1.8553 3.03796 −1.1457 1.54336 3.03796 1.21185 −1.8412 3.03796−1.1259 1.52733 3.03796 1.18985 −1.8235 3.03796 −1.102 1.50282 3.037961.16322 −1.8019 3.03796 −1.0751 1.46996 3.03796 1.13168 −1.776 3.03796−1.043 1.42517 3.03796 1.09498 −1.7456 3.03796 −1.0085 1.37191 3.037961.05179 −1.7093 3.03796 −0.9731 1.314 3.03796 1.00236 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0.92114 3.43056 0.98222 −1.459 3.43056−1.0657 1.00594 3.43056 1.03204 −1.5199 3.43056 −1.1 1.08359 3.430561.07656 −1.5746 3.43056 −1.1298 1.15394 3.43056 1.11565 −1.6229 3.43056−1.1551 1.21701 3.43056 1.14957 −1.6649 3.43056 −1.1766 1.27704 3.430561.17806 −1.7004 3.43056 −1.1922 1.3299 3.43056 1.20231 −1.7309 3.43056−1.2014 1.37138 3.43056 1.22271 −1.7566 3.43056 −1.2056 1.40516 3.430561.23927 −1.7776 3.43056 −1.2058 1.4306 3.43056 1.25266 −1.7947 3.43056−1.2026 1.44942 3.43056 1.26273 −1.8078 3.43056 −1.1982 1.45909 3.430561.26683 −1.8195 3.43056 −1.194 1.46439 3.43056 1.26617 −1.83 3.43056−1.1913 1.46651 3.43056 1.26286 −1.8378 3.43056 −1.1899 1.46731 3.43056−1.2038 1.32884 3.97156 1.22894 −1.832 3.97156 −1.203 1.32911 3.971561.2243 −1.8358 3.97156 −1.2015 1.32977 3.97156 1.21701 −1.8391 3.97156−1.1983 1.33043 3.97156 1.20694 −1.8399 3.97156 −1.1918 1.3303 3.971561.19555 −1.8359 3.97156 −1.182 1.32752 3.97156 1.18217 −1.8265 3.97156−1.1661 1.31851 3.97156 1.16481 −1.8141 3.97156 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5.53082 −0.2381 −0.1829 5.53082 −0.0368 −0.8575 5.53082−0.1731 −0.2711 5.53082 −0.1067 −0.7712 5.53082 −0.1073 −0.3588 5.53082−0.175 −0.6837 5.53082 −0.0408 −0.446 5.53082 −0.2422 −0.5953 5.530820.0265 −0.5327 5.53082 −0.3082 −0.506 5.53082 0.09434 −0.6188 5.53082−0.3727 −0.4157 5.53082 0.16284 −0.7046 5.53082 −0.4361 −0.3242 5.530820.23161 −0.7901 5.53082 −0.4981 −0.2319 5.53082 0.30064 −0.8754 5.53082−0.5586 −0.1383 5.53082 0.36994 −0.9605 5.53082 −0.6177 −0.0439 5.530820.43712 −1.0426 5.53082 −0.6736 0.04863 5.53082 0.50204 −1.1219 5.53082−0.7261 0.13899 5.53082 0.56458 −1.1983 5.53082 −0.7755 0.22711 5.530820.62487 −1.2719 5.53082 −0.8218 0.3127 5.53082 0.68291 −1.3426 5.53082−0.8648 0.39591 5.53082 0.73842 −1.4106 5.53082 −0.905 0.47634 5.530820.79142 −1.4759 5.53082 −0.9423 0.55398 5.53082 0.83979 −1.5357 5.53082−0.9768 0.62871 5.53082 0.88325 −1.5899 5.53082 −1.007 0.69722 5.530820.92194 −1.6385 5.53082 −1.0334 0.75936 5.53082 0.95599 −1.6816 5.53082−1.0559 0.81488 5.53082 0.98527 −1.7191 5.53082 −1.0755 0.86748 5.530821.00992 −1.7509 5.53082 −1.0905 0.91385 5.53082 1.03072 −1.7782 5.53082−1.1002 0.95003 5.53082 1.04821 −1.8012 5.53082 −1.1059 0.97957 5.530821.06239 −1.8203 5.53082 −1.108 1.00183 5.53082 1.07365 −1.8355 5.53082−1.1073 1.01866 5.53082 1.08226 −1.8475 5.53082 −1.1049 1.0278 5.530821.0861 −1.8578 5.53082 −1.102 1.03297 5.53082 1.08571 −1.8669 5.53082−1.0998 1.03496 5.53082 1.08306 −1.8738 5.53082 −1.0986 1.03589 5.53082

It is noted that the first column of TABLE I lists X coordinate valuesof the pressure side at each respective Z coordinate value of the thirdcolumn, the second column lists Y coordinate values of the pressure sideat each respective Z coordinate value of the third column, the fourthcolumn lists X coordinate values of the suction side at each respectiveZ coordinate value of the sixth column, and the fifth column lists Ycoordinate values of the suction side at each respective Z coordinatevalue of the sixth column. The Z coordinate values of the third columnand the sixth column are equal to each other for each respective row. Asset forth in TABLE I, at each respective Z coordinate value, the airfoilshape 110 of the airfoil 82 (e.g., a cross-sectional profile takenthrough line 6-6 of FIGS. 3, 4, and 5) is defined by multiple sets ofCartesian coordinate values of X, Y, and Z for both the suction side 62(e.g., suction side profile 112) and the pressure side 64 (e.g.,pressure side profile 114). For example, at each respective Z coordinatevalue, the suction side profile 112 of the suction side 62 may bedefined by at least equal to or greater than 10, 15, 20, 25, 30, 35, 40,45, 50, or 55 (e.g., 56) sets of Cartesian coordinate values of X, Y,and Z. Similarly, at each respective Z coordinate value, the pressureside profile 114 of the pressure side 64 may be defined by at leastequal to or greater than 10, 15, 20, 25, 30, 35, 40, 45, 50, or 55(e.g., 56) sets of Cartesian coordinate values of X, Y, and Z.Furthermore, in the Z direction along the Z axis 74, the airfoil profileof the airfoil 82 may be defined by multiple sets of Cartesiancoordinate values of X, Y, and Z at multiple Cartesian coordinate valuesof Z, such as at least equal to or greater than 2, 3, 4, 5, 6, 7, 8, 9,10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,28, 29, 30, or more Cartesian coordinate values of Z.

As noted above, the Cartesian coordinate values of X, Y, and Z as setforth in TABLE I may be non-dimensional values convertible todimensional distances. For example, the Cartesian coordinate values ofX, Y, and Z may be multiplied by a scale factor F (e.g., F equal to 1,greater than 1, or less than 1) to obtain the desired dimensionaldistances. In certain embodiments, a nominal airfoil profile of theairfoil 82 may be partially or entirely (e.g., all or part of thesuction side 62, all or part of the pressure side 64, or a combinationthereof) defined by the Cartesian coordinate values of X, Y, and Z asset forth in TABLE I. The nominal airfoil profile of the airfoil 82 alsomay be covered by the coating 120, which increases the X and Y values asset forth in TABLE I.

For example, the airfoil 82 may include a first suction portion (e.g.,80) of the nominal airfoil profile substantially in accordance withCartesian coordinate values of X, Y, and Z of the suction side 62 as setforth in TABLE I, wherein the X and Y values of the suction side 62 arecoordinate values that couple together (e.g., in a smooth continuousand/or curved manner) to define suction side sections of the firstsuction portion (e.g., 80) of the nominal airfoil profile at each Zcoordinate value, and the suction side sections of the first suctionportion (e.g., 80) of the nominal airfoil profile are coupled together(e.g., in a smooth continuous and/or curved manner) to define the firstsuction portion (e.g., 80). In such an embodiment, the airfoil 82 has anairfoil length L along the Z axis 74, and the first suction portion(e.g., 80) comprises a first portion length l along the Z axis asillustrated and described above with reference to FIG. 4. The firstportion length l is less than or equal to the airfoil length L. Thefirst portion length l may include greater than or equal to 2, 3, 4, 5,6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,25, 26, 27, 28, 29, 30, or more Cartesian coordinate values of Z (e.g.,consecutive values of Z) in TABLE I. The first portion length l may beequal to or greater than approximately 5, 10, 15, 20, 25, 30, 35, 40,45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 99, or 100 percent of theairfoil length L.

By further example, the airfoil 82 may include a second suction portion(e.g., 80) of the nominal airfoil profile substantially in accordancewith the Cartesian coordinate values of X, Y, and Z of the suction side62 as set forth in TABLE I, wherein the X and Y values of the suctionside 62 are coordinate values that couple together (e.g., in a smoothcontinuous and/or curved manner) to define suction side sections of thesecond suction portion (e.g., 80) of the nominal airfoil profile at eachZ coordinate value, the suction side sections of the second suctionportion (e.g., 80) of the nominal airfoil profile are coupled together(e.g., in a smooth continuous and/or curved manner) to define the secondsuction portion (e.g., 80). In such an embodiment, the second suctionportion (e.g., 80) has a second portion length l along the Z axis, thesecond portion length l is less than or equal to the airfoil length L,and the first and second suction portions (e.g., 80, 102, 104, 106) areoffset from one another along the Z axis as illustrated and describedabove with reference to FIGS. 4 and 5. Again, the second portion lengthl may include greater than or equal to 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,30, or more Cartesian coordinate values of Z (e.g., consecutive valuesof Z) in TABLE I. The second portion length l may be equal to or greaterthan approximately 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65,70, 75, 80, 85, 90, 95, 99, or 100 percent of the airfoil length L.Furthermore, the first and second suction portions (e.g., 80, 102, 104,106) may be separated by an offset distance of equal to or greater thanapproximately 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75,80, 85, or 90 percent of the airfoil length L.

By further example, the airfoil 82 may include a first pressure portion(e.g., 80) of the nominal airfoil profile substantially in accordancewith the Cartesian coordinate values of X, Y, and Z of the pressure side64 as set forth in TABLE I, wherein the X and Y values of the pressureside 64 are coordinate values that couple together (e.g., in a smoothcontinuous and/or curved manner) to define pressure side sections of thefirst pressure portion (e.g., 80) of the nominal airfoil profile at eachZ coordinate value, the pressure side sections of the first pressureportion (e.g., 80) of the nominal airfoil profile are coupled together(e.g., in a smooth continuous and/or curved manner) to define the firstpressure portion (e.g., 80). In such an embodiment, the first pressureportion (e.g., 80) comprises a second portion length l along the Z axis,and the second portion length l is less than or equal to the airfoillength L as illustrated and described above with reference to FIG. 4.Similar to the first suction portion (e.g., 80), the second portionlength l of the first pressure portion (e.g., 80) may include greaterthan or equal to 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, or more Cartesiancoordinate values of Z (e.g., consecutive values of Z) in TABLE I. Thesecond portion length l of the first pressure portion (e.g., 80) may beequal to or greater than approximately 5, 10, 15, 20, 25, 30, 35, 40,45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 99, or 100 percent of theairfoil length L. In certain embodiments, the first suction portion(e.g., 80) and the first pressure portion (e.g., 80) at least partiallyoverlap with one another along the Z axis.

For example, in certain embodiments, the first and second portionlengths l may be substantially the same, and the first and secondportion lengths l may start at a common distance d relative to the base60 of the airfoil 82 and extend toward the tip 68 of the airfoil 82.However, in some embodiments, the first and second portion lengths l maybe different from one another, the first and second portion lengths lmay start at different distances d relative to the base 60 of theairfoil 82, or a combination thereof. Additionally, the airfoil 82 mayinclude a second pressure portion (e.g., 80) of the nominal airfoilprofile substantially in accordance with the Cartesian coordinate valuesof X, Y, and Z of the pressure side 64 as set forth in TABLE I, whereinthe first and second pressure portions (e.g., 80, 102, 104, 106) areoffset from one another along the Z axis 74.

Technical effects of the disclosed embodiments include an airfoil havinga first suction portion of a nominal airfoil profile substantially inaccordance with Cartesian coordinate values of X, Y, and Z of a suctionside as set forth in TABLE I, wherein the X and Y values of the suctionside are coordinate values that couple together to define suction sidesections of the first suction portion of the nominal airfoil profile ateach Z coordinate value, the suction side sections of the first suctionportion of the nominal airfoil profile are coupled together to definethe first suction portion, the airfoil includes an airfoil length alonga Z axis, the first suction portion comprises a first portion lengthalong the Z axis, the first portion length is less than or equal to theairfoil length, and the Cartesian coordinate values of X, Y, and Z arenon-dimensional values convertible to dimensional distances (e.g.,instances in inches or mm).

This written description uses examples to disclose the subject matter,including the best mode, and also to enable any person skilled in theart to practice the subject matter, including making and using anydevices or systems and performing any incorporated methods. Thepatentable scope of the subject matter is defined by the claims, and mayinclude other examples that occur to those skilled in the art. Suchother examples are intended to be within the scope of the claims if theyhave structural elements that do not differ from the literal language ofthe claims, or if they include equivalent structural elements withinsubstantial differences from the literal language of the claims.

1. A system comprising: an airfoil comprising a first suction portion ofa nominal airfoil profile substantially in accordance with Cartesiancoordinate values of X, Y, and Z of a suction side as set forth in TABLEI to a maximum of three decimal places, wherein the X and Y values ofthe suction side are coordinate values that couple together to definesuction side sections of the first suction portion of the nominalairfoil profile at each Z coordinate value, the suction side sections ofthe first suction portion of the nominal airfoil profile are coupledtogether to define the first suction portion, the airfoil comprises anairfoil length along a Z axis, the first suction portion comprises afirst portion length along the Z axis, the first portion length is lessthan or equal to the airfoil length, and the Cartesian coordinate valuesof X, Y, and Z are non-dimensional values convertible to dimensionaldistances.
 2. The system of claim 1, wherein the airfoil comprises asecond suction portion of the nominal airfoil profile substantially inaccordance with the Cartesian coordinate values of X, Y, and Z of thesuction side as set forth in TABLE I to the maximum of three decimalplaces, wherein the X and Y values of the suction side are coordinatevalues that couple together to define suction side sections of thesecond suction portion of the nominal airfoil profile at each Zcoordinate value, the suction side sections of the second suctionportion of the nominal airfoil profile are coupled together to definethe second suction portion, the second suction portion comprises asecond portion length along the Z axis, the second portion length isless than or equal to the airfoil length, and the first and secondsuction portions are offset from one another along the Z axis.
 3. Thesystem of claim 1, wherein the airfoil comprises a first pressureportion of the nominal airfoil profile substantially in accordance withthe Cartesian coordinate values of X, Y, and Z of a pressure side as setforth in TABLE I to the maximum of three decimal places, wherein the Xand Y values of the pressure side are coordinate values that coupletogether to define pressure side sections of the first pressure portionof the nominal airfoil profile at each Z coordinate value, the pressureside sections of the first pressure portion of the nominal airfoilprofile are coupled together to define the first pressure portion, thefirst pressure portion comprises a second portion length along the Zaxis, and the second portion length is less than or equal to the airfoillength.
 4. The system of claim 3, wherein the first suction portion andthe first pressure portion at least partially overlap with one anotheralong the Z axis.
 5. The system of claim 4, wherein the first and secondportion lengths are substantially the same, and the first and secondportion lengths start at a common distance relative to a base of theairfoil and extend toward a tip of the airfoil.
 6. The system of claim1, wherein the first portion length of the first suction portion startsat a distance relative to a base of the airfoil and extends toward a tipof the airfoil.
 7. The system of claim 1, wherein the first portionlength includes greater than or equal to two consecutive Cartesiancoordinate values of Z in TABLE I.
 8. The system of claim 1, wherein thefirst portion length is equal to or greater than approximately 10percent of the airfoil length.
 9. The system of claim 1, wherein thefirst portion length is equal to or greater than approximately 25percent of the airfoil length.
 10. The system of claim 1, wherein thefirst portion length is equal to or greater than approximately 50percent of the airfoil length.
 11. The system of claim 1, wherein thefirst portion length is equal to or greater than approximately 75percent of the airfoil length.
 12. The system of claim 1, wherein thefirst portion length is equal to approximately 100 percent of theairfoil length.
 13. The system of claim 1, wherein the airfoil comprisesthe nominal airfoil profile substantially in accordance with Cartesiancoordinate values of X, Y, and Z as set forth in TABLE I to the maximumof three decimal places along an entirety of both the suction side and apressure side of the airfoil.
 14. The system of claim 1, wherein theairfoil comprises a coating disposed over the first suction portion ofthe nominal airfoil profile substantially in accordance with theCartesian coordinate values of X, Y, and Z of the suction side as setforth in TABLE Ito the maximum of three decimal places.
 15. The systemof claim 14, wherein the coating increases the X and Y values of thesuction side in TABLE I to the maximum of three decimal places by nogreater than approximately 3.5 mm along the first suction portion. 16.The system of claim 1, comprising a plurality of compressor airfoils ofa compressor stage, wherein each of the plurality of compressor airfoilscomprises the airfoil having the first suction portion of the nominalairfoil profile substantially in accordance with the Cartesiancoordinate values of X, Y, and Z of the suction side as set forth inTABLE Ito the maximum of three decimal places.
 17. The system of claim1, comprising a compressor having the airfoil.
 18. The system of claim17, comprising a gas turbine engine having the compressor, a combustor,and a turbine.
 19. The system of claim 1, wherein the airfoil is athirteenth stage compressor airfoil.
 20. The system of claim 19, whereinthe airfoil is a compressor rotor blade.
 21. The system of claim 1,wherein the airfoil comprises the first suction portion of the nominalairfoil profile substantially in accordance with Cartesian coordinatevalues of X, Y, and Z of the suction side as set forth in TABLE I to amaximum of four decimal places.
 22. The system of claim 1, wherein theairfoil comprises the first suction portion of the nominal airfoilprofile substantially in accordance with Cartesian coordinate values ofX, Y, and Z of the suction side as set forth in TABLE I to a maximum offive decimal places.
 23. A system comprising: an airfoil comprising asuction side of a nominal airfoil profile substantially in accordancewith Cartesian coordinate values of X, Y, and Z of the suction side asset forth in TABLE Ito a maximum of three decimal places, wherein the Xand Y values of the suction side are coordinate values that coupletogether to define suction side sections of the suction side of thenominal airfoil profile at each Z coordinate value, the suction sidesections of the suction side of the nominal airfoil profile are coupledtogether to define the suction side, and the Cartesian coordinate valuesof X, Y, and Z are non-dimensional values convertible to dimensionaldistances.
 24. A system comprising: an airfoil comprising a nominalairfoil profile substantially in accordance with Cartesian coordinatevalues of X, Y, and Z as set forth in TABLE I to a maximum of threedecimal places, wherein the X and Y values are coordinate values thatcouple together to define airfoil sections of the nominal airfoilprofile at each Z coordinate value, the airfoil sections of the nominalairfoil profile are coupled together to define an entirety of theairfoil, and the Cartesian coordinate values of X, Y, and Z arenon-dimensional values convertible to dimensional distances.